An optimal boundary control approach to the Cherrier-Escobar problem

TL;DR

This paper develops an optimal boundary control framework for the boundary obstacle problem on Riemannian manifolds, linking control solutions to geometric properties like scalar curvature and mean curvature, with explicit results on the standard ball.

Contribution

It introduces a novel boundary control approach for the Cherrier-Escobar problem, establishing conditions for optimal controls and their geometric implications, including sharp inequalities and uniqueness results.

Findings

Optimal controls coincide with their states when the Cherrier-Escobar invariant is positive.

Optimal controls minimize the Cherrier-Escobar functional, producing conformal metrics with desired curvature properties.

For the unit ball, the standard bubbles are the unique optimal controls, satisfying a sharp Sobolev trace inequality.

Abstract

We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the Cherrier-Escobar invariant of the compact Riemannian manifold with boundary is positive, we show that the optimal controls are equal to their associated optimal states. Moreover, we show that the optimal controls are minimizers of the Cherrier-Escobar functional, and hence induce conformal metrics with zero scalar curvature and constant mean curvature. Furthermore, we show the existence of an optimal control under an Aubin type assumption. For the standard unit ball, we derive a sharp Sobolev trace type inequality and prove that the standard bubbles-namely conformal factor of metrics conformal to the standard one with zero scalar curvature and constant mean curvature -- are the only optimal controls and hence equal to their associated optimal states.